Calculus I question (abs and loc max/min).?

Question

Find the absolute and local maximum and minimum values of f(x) = 8sinx over [-2pi,2pi].

Answer 1

Function over the interval .

If f be defined on an interval I containing c.

Then, f(c) is the absolute minimum of f on I if      for all x in I.

and f(c) is the absolute maximum of f on I if         for all x in I.

The value f(c) is called local minimum or local maximum at critical numbers only.

Critical numbers can be found by differentiating the function and setting f'(x) equal to zero.

Set       .

        

           

In the interval , is zero when .

These are the critical numbers.

By evaluating f at these four critical numbers and at the end points of the interval,

you can find the absolute extrema and local extrema.

.

The minimum and maximum values in the entire interval are absolute extrema.

Absolute maximum is at , .

Absolute minimum is at , .

The minimum and maximum values at the critical points are local extrema.

Local maximum is at   , .

Local minimum is at   , .